I always amend the soil with some kind of soil amendment..I likr Gardners Bloom Organic planting mix.I don't use BLOOD & BONE meal because the racoons around here like it and will dig it up.
Yes you can do it now and water right after transplanting.Be carefull not to plant grasses too deep.At or just above grade.

I am going to transplant some Mondo Grass. Is this a good time of year to do it? Should I put any bone meal or anything else in the soil to help it get going? Thanks

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Are you going to divide and separate it when you transplant it? What are your intentions or purpose of transplanting?

The reason I am asking is that Mondo is usually so root bound or rather intertwined, that you can generally transplant the thing like carpet. Also, while it is beneficial to ammend the soil when you plant, Mondo is a very tough perennial ground cover and can tolerate just transplanting into fresh topsoil. Just be sure to water and apply a slow release fert.